Principles of Mathematics in Operations Research

3.1 Inner Products 37

Proposition 3.1.16 The cosine of the angle between any two vectors u and

v is

COSC :

U V T

iu \\v\\

Remark 3.1.17 The law of cosines:

\\u — v\\ = ||u|| + \\v|| — 2 ||u|| llvll cose.

3.1.4 Projection

Let p = xv where W = x 6 R is the scale factor. See Figure 3.3.

(u - p) J- v «=> vT(u — p) = 0 & x =

VxT U

X-Axis

Fig. 3.3. Projection

Definition 3.1.18 The projection p of the vector u onto the line spanned by
T
the vector v is given by p — %^-v.
The distance from the vector u to the line is (Schwartz inequality) therefore

ir u

u =-v

v^1 v

u

T

u - 2^- + &?v

T

v = (»

r

»)("

r

;)-(«

r

«)

a

,

V^1 V V^1 V V^1 V

3.1.5 Symmetric Matrices

Definition 3.1.19 A square matrix A is called symmetric if AT = A.

Proposition 3.1.20 Let A e Rmxn, rank{A) = r. The product ATA is a
symmetric matrix and rank(ATA) = r.